What is the best way to change edge weights on a mesh?
You could use a skin modifier to create the outline. You can adjust the thickness of the skinning with &#9096; CtrlA in edit mode. The properties region N can also be used to adjust the radius.



Make a duplicate of your mesh to use for the outline, in the duplicate delete all the faces and the edges that cross over the pathways so you only have edges where you want the outline to be. Add the skin modifier to this mesh. For each disconnected piece of mesh you will need to mark a vertex as root (use the button in the modifier), otherwise that section of mesh will not be skinned.



Instead of the skin modifier, another option is to convert the outline mesh to a curve object and use the curves bevel depth to adjust the outline thickness.